The value of the integral I=∫_-1^1≤ft(x+x^3+x^5)dx is :

The correct answer is Option (3) → 0

Let \(f(x) = x + x^3 + x^5\). Every power is odd, so

    \(f(-x) = -x - x^3 - x^5 = -f(x)\)

and f is an odd function. For an odd function the integral over a symmetric interval [−a, a] is zero, because the area on the left of the y-axis cancels the area on the right:

    \( \displaystyle \int_{-a}^{a} f(x)\,dx = 0 \)

Directly:

    \( \displaystyle I = \left[\frac{x^2}{2} + \frac{x^4}{4} + \frac{x^6}{6}\right]_{-1}^{1} \)

    \( \displaystyle = \left(\frac{1}{2} + \frac{1}{4} + \frac{1}{6}\right) - \left(\frac{1}{2} + \frac{1}{4} + \frac{1}{6}\right) = 0 \)